Atrayonis suggested he and I write out our respective (and probably starkly divergent) ideas on the future of Morrowind after the events of TES: III (including TR events of course). I've decided to make a start at that, but will divide my write-up into at least two if not more parts, this being the first.
While I do incorporate some Bethesda lore of the later games into my account, though often with changes, that's more a matter of me finding it interesting to incorporate those ideas. My concept does diverge from later Bethesda lore quite heavily in areas, especially with Skyrim's Dragonborn expansion, which after all deals with Morrowind's later history. As all those divergences will not happen within TR's gameplay, though, that's pretty irrelevant, and this is really more of a thought exercise.
First thing is to establish the state of things at the end of Morrowind. This can vary a bit depending on what questlines the player completes, and perhaps how the player completes them. Also, it's possible plans for the questlines will change later, so this isn't really set in stone. In no particular order:
1. Two or all three of the Tribunal are dead, the dissident priests are no longer persecuted lending relevance to their message and to Ashlander belief -- all the more so with the appearance of a Nerevarine recognized by, at least, a majority of the houses and Ashlander tribes. While the Temple hasn't been overthrown, it's clearly only a matter of time until faith in it dies completely, and it is in disarray. It certainly has lost its ability to regulate and unite the activity of the great houses.
2. House Indoril has either succeeded in repelling House Hlaalu in the Thirr River Conflict, or has lost and is continuing to lose its western territory. Even at best, its 'victory' is pyrrhic, revealing deep divisions within a house that had prided itself in its unity and common mission. Much of its leadership is dead, and in the case of 'victory' an outlander upstart now sits as their helm.
3. House Redoran has managed to quell an uprising of local Nords, but the uprising has awoken slumbering tensions in a land that had never been able to maintain isolation from outlanders like the eastern houses did. The sovereignty of House Redoran is shaken by runaway Imperialization, Hlaalu are nipping at their heels, especially on Vvardenfell, and the other houses are too preoccupied or indifferent to provide any aid. Skyrim's High King Thian, who supported the uprising, is still going strong. Bolvyn Venim's suicide mission against Dagoth Ur ends with Venim dead at the hands of the Nerevarine, who kills Dagoth Ur rendering the mission irrelevant.
4. While House Hlaalu's expansion on Vvardenfell and across the Thirr has possibly been stalled by the activities of House Redoran and House Indoril respectively, they are still in a very strong position. After a tense moment facing the prospect of fighting a coalition of the other great houses (except Telvanni probably) unaided, Hlaalu emerge with an alliance with House Dres, arguably the next strongest house at that point, and the cooperation of King Hlaalu Helseth. (While a member of House Hlaalu, Helseth is a political outsider who was not by default aligned with his house). They appear to be unstoppable.
5. House Telvanni is continuing to take advantage of the political chaos in Morrowind to spread outwards, as upstarts repelled by the ancient wizards in the East build up power beyond their reach. Telvanni's claim to being a unified house has always been tenuous, but with the absent Archmagister Rilvin Dral and divisive de facto Archmagister Gothren replaced by an outlander Hortator and Nerevarine, House Telvanni seems poised to strike out in a new and unpredictable direction. At the same time, though, several of those ancient wizards die in the process, and while that provides fertile soil for new growth, it also carries uncertainty with it.
6. House Dres has been secretely struggling with creditors of major deals they made at the very beginning of Tribunal history. While managing to keep their secret struggle hidden from the other houses -- maintaining their strength for the moment as Morrowind's second strongest house and allying with the strongest, House Hlaalu -- internally House Dres has cracked. The secrets are beginning to leak, the Matriarchs who secured the deals, uniting the squabbling Dres clans in a confederacy and guiding them forward in the process, are either all or mostly dead. While the Sload have been dealt with, the Hist are still looming, and its unclear how much longer the Dres will keep supplying the Hist with Dunmer as their part of that deal, if they haven't already stopped.
7. While fortunes have largely favoured the endeavours of the Empire in Morrowind -- supportive House Hlaalu becoming the dominant local power in the province, the reformer Helseth exerting and expanding his power as king, the Tribunal weakened and possibly killed and the Temple on the way out with them, and one of the Empire's agents being recognized as Nerevarine and Hortator by the majority of local factions -- the Empire is not able to savour it due to internal instability. Garrisons in hostile eastern Morrowind have been abandoned in favour of stronger defences in the West, and the Empire's Legions are being recalled.